Definitely possible (I just did it). Definitely not easy (in my experience).
I just opened an HSBC account in the UK without being a UK resident. I am a UK citizen, however - but reside in the US (dual US/UK citizen). I opened it with zero deposit.
I was in UK and needed to open an account to receive an inheritance. I walked into local branches of Lloyds, HSBC, NatWest - they all said it was not possible. Barclays referred me to their international department (online). Lloyds tried hard to help, but even though they could open an account for a non-resident, they said they couldn't do it for a US resident due to some 'licensing' issue specific to the US. NatWest were simply useless. HSBC, while being somewhat useless, did take my phone number and I did get a call back the next day from a regional person who said it was possible. He contacted the branch manager of a branch he was familiar with and set up an appointment for me (about a week later). After a 2-hour ordeal, I was pretty much set up; I got the sort code and account number about 3 days later. The account has no minimums to avoid fees; very much a no-frills basic account. This is NOT an HSBC International account or US account; it's a standard UK account.
Barclays international, based in the Isle of Man, also were very helpful. They set up a phone and video appointment, and did everything 'online' - including document verification, by holding documents up in front of the laptop video camera! Quite 'modern'. However, over a month later they have not confirmed the opening of the account and they do have something like a 25k GBP minimum to avoid fees (I chose to open both the HSBC and Barclays accounts in case one fell through for any reason).
Overall, I found the in-branch staff to be somewhat useless, and the online 'chat' help equally useless. I spent hours 'chatting' with an HSBC support rep all to no avail. It's obviously an area where they get very little exposure.